# ROC Curve Horizontal Line

I'm trying to conceptually understand 'why' the horizontal gray line in ROC curve below would be considered random.

I know the basics of the ROC in that the farther the curved black line is from the horizontal the better, and that the screenshot below (the random point highlighted) is stating that a prediction threshold of .473 will result in .533 sensitivity and .785 specificity. With sensitivity being the true positive and specificity being the true negative.

That being said, I'm missing something basic and can't grasp why .6 sensitivity with .4 specificity would be the 'random' result. Hoping someone can help me grasp it in simple, layman's terms. Thank's in advance!

Bonus question.. sometimes I see Specificity along the x axis and sometimes it's the false positive rate instead. Not sure why you would use one vs the other or the meaning in that.

• You have two classes: 0,1. What is the specificity/sensitivity of a coin flip where 1 comes up with probably $p$ and 0 with $1-p$? Now trace out these pairs for all $p\in[0,1]$. Commented Oct 9, 2019 at 1:59